On a Family of Functions Defined Over Sums of Primes
Department of Mathematics
Let r and m be real numbers so that the sum
∑p ≤ x pr
logm p diverges
as x → ∞. Here p runs
over all primes not exceeding x.
In this paper, we give an
asymptotic formula for each
as x → ∞. The case where x
is the nth prime
number is of particular interest.
Here we use a method developed by Salvy to give an
asymptotic formula for
as n → ∞,
which generalizes, for instance, the previously
known one for
the sum of the first n prime numbers.
Full version: pdf,
(Concerned with sequences
Received December 17 2018; revised versions received December 19 2018; March 29 2019;
July 8 2019.
Published in Journal of Integer Sequences,
August 23 2019.
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